Let \(\mathcal{C}\) be the collection of all functions \(j : \{ 1,\dots, m\} \to \{0, 1\}\) with \(m \geq 2\). Each function \(j\) can be identified with the corner of an \(m\)-dimensional cube \([0,1]^{m}\). For \(j \in \mathcal{C}\), it is defined the projection \(\Pi_j : \mathbb{R}^{2m} \to \mathbb{R}^{m}\) by \(\Pi_j (x_1^{0},\dots, x_m^{0},x_1^{1},\dots, x_m^{1})^{T} = (x_1^{j(1)},\dots, x_m^{j(m)})^{T}\). With this cubical structure, the authors consider the following singular Brascamp-Lieb form \[ p.v. \int_{\mathbb{R}^{2m}} ( \prod_{j \in \mathcal{C}} F_j(\Pi_j x) ) \, K(\Pi x) \, dx, \] where each \(F_j : \mathbb{R}^{m} \to \mathbb{R}\) belongs to \(C_{c}^{\infty}(\mathbb{R}^{m})\), the \(\Pi_j\)’s are as above, \(K\) is a Calderón-Zygmund kernel in \(\mathbb{R}^{m}\) of order \(2^{6m}\), and \(\Pi : \mathbb{R}^{2m} \to \mathbb{R}^{m}\) is a surjective linear map such that for each \(j \in \mathcal{C}\) the composition \(\Pi_j \Pi^{T}\) satisfies certain regularity condition. The main result of this paper is the following: for \(2^{m-1} < p_j \leq \infty\) with \(\sum_{j \in \mathcal{C}} \frac{1}{p_j} = 1\), the authors using the sparse domination technique and local bounds prove that \[ \left| p.v. \int_{\mathbb{R}^{2m}} ( \prod_{j \in \mathcal{C}} F_j(\Pi_j x) ) \, K(\Pi x) \, dx \right| \leq C \prod_{j \in \mathcal{C}} \| F_j \|_{p_j}, \] where \(C\) is a positive constant independent of the \(F_j\)’s. This result generalizes Theorem 1 in [P. Durcik and C. Thiele, Bull. Lond. Math. Soc. 52, No. 2, 283–298 (2020; Zbl 1442.26025)]. Moreover, the threshold \(2^{m-1}\) is sharp.